Since \(\gamma _A\in \Gamma _H^+\) , while \(\gamma _H\)  minimizes \(\Vert \nu \Vert ^2\)  over \(\nu \in \Gamma _H^+\) , we derive (REF ) from [1]}.
Let now Frostman's maximum principle be satisfied. Noting that then \(\kappa \gamma _A=1\)  holds n.e. on each \(K\in \mathfrak {C}_H\) , hence \(\gamma _K\) -a.e. (Lemma REF ), whereas
\(\gamma _K\rightarrow \gamma _H\text{ \ strongly in $\mathcal {E}^+$ as $K\uparrow H$},\) 
we obtain
\(\kappa (\gamma _A,\gamma _H)=\lim _{K\uparrow H}\,\kappa (\gamma _A,\gamma _K)=\lim _{K\uparrow H}\,\int \kappa \gamma _A\,d\gamma _K=\lim _{K\uparrow H}\,\gamma _K(X)=\lim _{K\uparrow H}\,\Vert \gamma _K\Vert ^2=\Vert \gamma _H\Vert ^2,\) 

whence (REF ). (The forth equality here is valid by (REF ) with \(A:=K\) , while the first follows from (REF ) with the aid of the Cauchy–Schwarz inequality.)
